US 7058676 B2 Abstract In order to transform an input digital signal (x
_{n}) into one or more output digital signals (y_{n}) containing even-indexed samples (y_{2n}) and odd-indexed samples (y_{2i+n}), this filtering method includes at least one iteration which contains an operation of modifying even-indexed samples (y_{2n}) by a function (R) of weighted odd-indexed samples (y_{2n+1}), and an operation of modifying odd-indexed samples (y_{2n+1}) by a function (R) of weighted even-indexed samples (β_{0,j}(y_{2n}−y_{2n+2})). The weighted samples are obtained by at least one weighting operation. At least one of the weighting operations is applied to the difference between two consecutive even-indexed samples.Claims(63) 1. A filtering method for transforming an input digital signal into one or more output digital signals having even-indexed samples and odd-indexed samples, said method including at least one iteration comprising the steps of:
modifying the even-indexed samples by a function of weighted odd-indexed samples; and
modifying the odd-indexed samples by a function of weighted even-indexed samples,
wherein the weighted samples are obtained by at least one weighting operation applied to a difference between two consecutive even-indexed samples.
2. A filtering method according to
3. A filtering method according to
weighting, by a first weighting coefficient, at least one odd-indexed sample adjacent to an even-indexed sample currently being modified, so as to obtain a weighted odd-indexed sample;
modifying at least one even-indexed sample using the at least one weighted odd-indexed sample;
weighting, by a second weighting coefficient, even-indexed samples adjacent to an odd-indexed sample currently being modified, so as to obtain weighted even-indexed samples; and
modifying at least one odd-indexed sample using at least one weighted even-indexed sample.
4. A filtering method according to
5. A filtering method according to
where α
_{0,j }designates the first weighting coefficient, β_{0,j }designates the second weighting coefficient, i and j are integers, m_{j }is a value defined by a recurrence m_{0}=(−1)^{L} _{0 }and m_{j}=−m_{j−1}, and L_{0 }is a predetermined integer.6. A filtering method according to
7. A filtering method according to
8. A filtering method according to
γ=−1/(2β _{0, L} _{ 0 } _{−1})where γ designates the third weighting coefficient, L
_{0 }is a predetermined parameter and β_{0, L} _{ 0 } _{−1 }designates the weighting coefficient used in the preceding step.9. A filtering method according to
10. A filtering method for transforming one or more input digital signals into an output digital signal, the input signals including even-indexed samples and odd-indexed samples, said method including at least one iteration comprising the steps of:
modifying odd-indexed samples by a function of weighted even-indexed samples; and
modifying even-indexed samples by a function of weighted odd-indexed samples,
wherein the weighted samples are obtained by at least one weighting operation applied to a difference between two consecutive even-indexed samples.
11. A filtering method according to
12. A filtering method according to
weighting, by a first weighting coefficient, even-indexed samples adjacent to an odd sample currently being modified, so as to obtain weighted even-indexed samples;
modifying at least one odd-indexed sample using at least one weighted even-indexed sample;
weighting, by a second coefficient, at least one odd-indexed sample adjacent to an even sample currently being modified, so as to obtain a weighted odd-indexed sample; and
modifying at least one even-indexed sample using at least one weighted odd-indexed sample.
13. A filtering method according to
14. A filtering method according to
where α
_{0,j }designates the second weighting coefficient, β_{0,j }designates the first weighting coefficient, i and j are integers, m_{j }is a value defined by a recurrence m_{0}=(−1)^{L} _{0 }and m_{j}=−m_{j−1}, and L_{0 }is a predetermined integer.15. A filtering method according to
16. A filtering method according to
17. A filtering method according to
γ=−1/(2β _{0, L} _{ 0 } _{−1})where γ designates the third weighting coefficient, L
_{0 }is a predetermined parameter and β_{0, L} _{ 0 } _{−1 }designates the weighting coefficient used in the following step.18. A filtering method according to
19. A filtering method according to
20. A filtering method according to
21. A filtering method according to
22. A filtering method according to
23. A filtering method according to
24. A filtering method according to
25. A signal processing device, comprising means adapted to implement a filtering method according to
26. A digital filtering device adapted to transform an input digital signal into one or more output digital signals containing even-indexed samples and odd-indexed samples, said filtering device comprising:
at least one weighting module; and
means for modifying even-indexed samples by a function of weighted odd-indexed samples,
wherein weighted samples are supplied by said at least one weighting module, said modification means functioning iteratively, so as to modify even-indexed samples at least once and then odd-indexed samples at least once, and said at least one weighting module receives as an input the difference between two consecutive even-indexed samples.
27. A digital filtering device according to
28. A digital filtering device according to
means for weighting, by a first weighting coefficient, at least one odd-indexed sample adjacent to an even sample currently being modified, so as to obtain a weighted odd-indexed sample;
means for modifying at least one even-indexed sample using at least one weighted odd-indexed sample;
means for weighting, by a second weighting coefficient, even-indexed samples adjacent to an odd sample currently being modified, so as to obtain weighted even-indexed samples; and
means for modifying at least one odd-indexed sample using the at least one weighted even-indexed sample.
29. A digital filtering device according to
30. A digital filtering device according to
where α
_{0,j }designates the first weighting coefficient, β_{0,j }designates the second weighting coefficient, i an j are integers, m_{j }is a value defined by a recurrence of m_{0}=(−1)^{L} _{0 }and m_{j}=−m_{j−1}, and L_{0 }is a predetermined integer.31. A digital filtering device according to
32. A digital filtering device according to
33. A digital filtering device according to
γ=−1/(2β _{0, L} _{ 0 } _{−1})where γ designates the third weighting coefficient, L
_{0 }is a predetermined parameter and β_{0, L} _{ 0 } _{−1 }designates the weighting coefficient used upstream of said additional filtering means.34. A digital filtering device according to
35. A digital filtering device adapted to transform one or more input digital signals into an output digital signal, the input signals containing even-indexed samples and odd-indexed samples, said filtering device comprising:
at least one weighting means;
means for modifying odd-indexed samples by a function of weighted even-indexed samples; and
means for modifying even-indexed samples by a function of weighted odd-indexed samples,
wherein said weighted samples are supplied by said at least one weighting means, said modification means functions iteratively, so as to modify odd-indexed samples at least once and then even-indexed samples at least once, and
wherein said at least one weighting means receives as an input the difference between two consecutive even-indexed samples.
36. A digital filtering device according to
37. A digital filtering device according to
means for weighting, by a first weighting coefficient, even-indexed samples adjacent to an odd sample currently being modified, so as to obtain weighted even-indexed samples;
means for modifying at least one odd-indexed sample using at least one weighted even-indexed sample;
means for weighting, by a second coefficient, at least one odd-indexed sample adjacent to an even sample currently being modified, so as to obtain a weighted odd-indexed sample; and
means for modifying at least one even-indexed sample using at least one weighted odd-indexed sample.
38. A digital filtering device according to
39. A digital filtering device according to
where α
_{0,j }designates the second weighting coefficient, β_{0,j }designates the first weighting coefficient, i and j are integers, m_{j }is a value defined by a recurrence m_{0}=(−1)^{L} _{0 }and m_{j}=−m_{j−1}, and L_{0 }is a predetermined integer.40. A digital filtering device according to
41. A digital filtering device according to
42. A digital filtering device according to
γ=−1/(2β _{0, L} _{ 0 } _{−1})where γ designates the third weighting coefficient, L
_{0 }is a predetermined parameter and β_{0, L} _{ 0 } _{−1 }designates the weighting coefficient used downstream of said additional filtering means.43. A digital filtering device according to
44. A digital filtering device according to
45. A digital filtering device according to
46. A digital filtering device according to
47. A digital filtering device according to
48. A digital filtering device according to claim
44, wherein the approximation function is a function of a real variable which supplies the first integer above said real variable.49. A digital filtering device according to
50. A signal processing device comprising a digital filtering device according to
51. A signal processing device comprising at least two digital filtering devices according to
52. A digital apparatus comprising a signal processing device according to
53. A digital photographic apparatus comprising a signal processing device according to
54. An encoding method comprising steps adapted to implement a filtering method according to
55. An encoding device comprising at least one filtering device according to
56. A digital compression method comprising steps adapted to implement a filtering method according to
57. A digital signal compression device comprising at least one filtering device according to
58. An information storage means which can be read by a computer or by a microprocessor, and which stores a program, comprising means adapted to implement a filtering method according to
59. A computer program product comprising code for implementing a filtering method according to
60. A digital apparatus comprising a signal processing device according to
61. A digital photographic apparatus comprising a signal processing device according to
62. A digital apparatus comprising a signal processing device according to
63. A digital photographic apparatus comprising a signal processing device according to
Description 1. Field of the Invention The present invention relates to digital filtering methods and devices. It finds an advantageous application to the filtering of digital images, in particular for the compression of images in accordance with “JPEG 2000”, currently being drafted. 2. Description of the Related Art Two techniques are notably known for implementing the filtering of digital signals using wavelet filters: filtering by convolution, and filtering by “lifting”. A few elementary concepts on wavelet filters, and then on lifting, are given below. It is known that the wavelet filtering of a monodimensional signal X the convolution of a low-pass filter H the convolution of a high-pass filter H
The result of the filtering, referred to as a wavelet filtering operation (H The low-pass filter and the high-pass filter must satisfy certain conditions, referred to as perfect reconstruction conditions, so that the perfect reconstruction of the signal X If the notation of the z transform is used for the finite impulse response filters, namely It is known that some wavelet filters have symmetry properties. An odd-length symmetric wavelet filter H(k) or WSS (Whole-Sample Symmetric) filter is defined by the following equation, referred to as WSS symmetry, between all the coefficients of the filter:
An even-length symmetric wavelet filter or HSS (Half-Sample Symmetric) filter is defined by the following equation, referred to as HSS symmetry, between all the coefficients of the filter:
An even-length antisymmetric wavelet filter or HSA (Half-Sample Antisymmetric) filter is defined by the following equation, referred to as HSA symmetry, between all the coefficients of the filter:
The technique of lifting, mentioned above, consists of implementing a filtering step by decomposing it into an equivalent sequence of elementary filtering operations, referred to as lifting steps. This decomposition is referred to as lifting factorisation. For more details on lifting factorisation, reference can usefully be made to the article by Ingrid DAUBECHIES and Wim SWELDENS entitled “ It should be stated in particular that, when a filtering operation is performed in the form of lifting steps, in order to transform a signal X There are two types of lifting step: high-pass lifting steps, and low-pass lifting steps. A high-pass lifting step consists of modifying odd-indexed samples by adding to them a sample function of even-indexed samples which are weighted or filtered:
The function R is, in general terms, any approximation of the variable x. The function R can be a rounding operator which rounds a real value x to an integer (such as the closest integer), or can simply be identity: R(x)=x. A low-pass lifting step consists of modifying even-indexed samples (low-pass) by adding to them a function of odd-indexed samples which are weighted or filtered:
In the case where R(x)=x, at each lifting step, each low-pass sample Y It is said that each low-pass lifting step lifts a low-pass filter (one which is associated with the previous low-pass lifting step) into another low-pass filter (the one which is associated with the current low-pass lifting step). The term lifting accounts for the fact that the support of the resulting low-pass filter (that is to say the number of coefficients of the filter) is greater than that of the corresponding low-pass filter at the previous step. The low-pass filter with the larger support finally obtained by lifting is called the “equivalent low-pass filter”. The same applies to the high-pass filters and the lifting steps, that is to say each high-pass lifting step can be seen as the lifting of a high-pass filter into a high-pass filter with a larger support, referred to as the “equivalent high-pass filter”. The same terminology is used in the case where the approximation function R is not identity. Thus the lifting-based implementation of a wavelet filtering operation (H Several types of symmetry are also encountered in the lifting steps: -
- there are two types of symmetric lifting steps:
- lifting steps with WSS symmetry, and
- lifting steps with HSS symmetry; and
- there are two types of antisymmetric lifting steps:
- lifting steps with WSA antisymmetry, and
- lifting steps with HSA antisymmetry.
- there are two types of symmetric lifting steps:
An odd-length symmetric lifting or WSS lifting (“Whole-Sample Symmetric lifting”) step is defined by a WSS symmetry relationship between the filtering coefficients of the lifting step:
An even-length symmetric lifting or HSS lifting (“Half-Sample Symmetric lifting”) step is defined by an HSS symmetry relationship between the filtering coefficients of the lifting step:
An odd-length antisymmetric lifting or WSA lifting (“Whole-Sample Antisymmetric lifting”) step is defined by a WSA symmetry relationship between the filtering coefficients of the lifting step:
An even-length antisymmetric lifting or HSA lifting (“Half-Sample Antisymmetric lifting”) step is defined by an HSA symmetry relationship between the filtering coefficients of the lifting step:
A single-coefficient lifting or SC lifting (“Single-Coefficient lifting”) step is defined by the fact that all the filtering coefficients of the lifting step except one (denoted a For the low-pass lifting steps, if k′=0, the lifting step is called a Right-side Single Coefficient low-pass lifting or RSC low-pass lifting step, and if k′=−1, the lifting step is called a Left-side Single Coefficient low-pass lifting or LSC low-pass lifting step. In a similar manner, for the high-pass lifting steps, if k′=1, the lifting step is called a Right-side Single Coefficient high-pass lifting or RSC high-pass lifting step, and if k′=0, the lifting step is called a Left-side Single Coefficient high-pass lifting or LSC high-pass lifting step. Given that, at the present time, more and more signal compression algorithms use wavelet filtering operations for the decorrelation of the signals, it is very interesting to have available an effective implementation of these filtering operations. Various image compression algorithms use bidimensional wavelet transformations, which consist of the successive application of monodimensional transformations of the type described in the introduction. An effective implementation of wavelet-based monodimensional filtering is lifting-based filtering. It is known in fact that lifting-based filtering reduces the number of filtering operations (multiplications and additions) and also affords a reversible implementation of wavelet-based filtering, that is to say with perfect reconstruction of the signal, without loss of information, in the case of signals whose sample values are integer. In addition, the step which is the reverse of a lifting step is obtained in a trivial fashion by changing the sign before the approximation function R given in equations (8) and (9) above. This makes it possible notably to use the same circuitry in a hardware implantation corresponding to both forward and reverse wavelet transformations. For a given pair of wavelet filters (H Two families of wavelet filters are known which have good signal decorrelation properties: the family WSS/WSS, abbreviated to WSS, and the family HSS/HSA, the notation A/B meaning that the low-pass filter H The problem consisting of implementing WSS wavelet filters by using HSS lifting steps has been resolved and is described in the article by Ingrid DAUBECHIES and Wim SWELDENS cited above. The problem consisting of implementing HSS/HSA wavelet filters has been partly resolved, by using WSA lifting steps. However, this solution is only partial since this implementation of the prior art applies only to a limited set of HSS/HSA wavelet filters and not to all of these filters. Moreover, the publication by E. MAJANI entitled “ The purpose of the present invention is to remedy this drawback. It proposes an implementation by lifting of all the HSS/HSA wavelet filters which cannot be implemented by using solely WSA lifting steps. It therefore forms a solution to the problem of the implementation by lifting of pairs of filters (H To this end, the present invention proposes a filtering method adapted to transform an input digital signal into one or more output digital signals having even-indexed samples and odd-indexed samples, this method including at least one iteration which contains an operation of modifying even-indexed samples by a function of weighted odd-indexed samples, an operation of modifying odd-indexed samples by a function of weighted even-indexed samples, the weighted samples being obtained by at least one weighting operation, this method being remarkable in that at least one of the weighting operations is applied to the difference between two consecutive even-indexed samples. Thus, the present invention allows notably the implementation by lifting of any pairs of HSS/HSA wavelet filters, and the implementation by lifting of any pair of orthogonal filters and the implementation by lifting of any pair of any wavelet filters with equal support. The invention also makes it possible, by virtue of the technique of filtering by lifting, to reduce the number of filtering operations (multiplications and additions) compared with a conventional convolution filtering and also guarantees the reversible character of the wavelet transformations, that is to say a perfect reconstruction of the signal, in the case of signals whose sample values are integer. According to a particular characteristic, the operation of modifying odd-indexed samples is performed following the operation of modifying even-indexed samples. In a particular embodiment, the aforementioned iteration consists notably of: weighting, by means of a first weighting coefficient, at least one odd-indexed sample adjacent to an even sample currently being modified, so as to obtain a weighted odd-indexed sample, modifying at least one even-indexed sample using at least one weighted odd-indexed sample, weighting, by means of a second weighting coefficient, even-indexed samples adjacent to an odd sample currently being modified, so as to obtain weighted even-indexed samples, and modifying at least one odd-indexed sample using at least one weighted even-indexed sample. “Adjacent” sample means a sample whose rank is consecutive with the sample in question. According to a particular characteristic, the second weighting coefficient is a function of the first weighting coefficient. The second weighting coefficient can for example depend on the first weighting coefficient as follows: According to a particular characteristic, at each iteration, the odd-indexed sample adjacent to the even sample currently being modified is alternately the sample of rank immediately lower or immediately higher. According to a particular characteristic, the filtering method includes, at the end of the aforementioned iteration, an additional filtering step including an operation of weighting by means of a third weighting coefficient. According to a particular characteristic, the third weighting coefficient is a function of the weighting coefficient used at the previous step, as follows:
In the application of the invention to the filtering of digital images, the input digital signal represents an image. According to a second aspect, the present invention also proposes a filtering method adapted to transform one or more input digital signals into an output digital signal, the input signals including even-indexed samples and odd-indexed samples, the method including at least one iteration which contains an operation of modifying odd-indexed samples by means of a function of weighted even-indexed samples, an operation of modifying even-indexed samples by means of a function of weighted odd-indexed samples, the weighted samples being obtained by means of at least one weighting operation, this method being remarkable in that at least one of the weighting operations is applied to the difference between two consecutive even-indexed samples. According to a particular characteristic, the operation of modifying even-indexed samples is performed following the operation of modifying odd-indexed samples. In a particular embodiment, the aforementioned iteration consists notably of: weighting, by means of a fourth weighting coefficient, even-indexed samples adjacent to an odd sample currently being modified, so as to obtain weighted even-indexed samples, modifying at least one odd-indexed sample using at least one weighted even-indexed sample, weighting, by means of a fifth weighting coefficient, at least one odd-indexed sample adjacent to an even sample currently being modified, so as to obtain a weighted odd-indexed sample, and modifying at least one even-indexed sample using at least one weighted odd-indexed sample. According to a particular characteristic, the fourth weighting coefficient is a function of the fifth weighting coefficient. The fourth weighting coefficient can for example depend on the fifth weighting coefficient as follows: According to the second aspect of the invention, according to a particular characteristic, at each iteration, the odd-indexed sample adjacent to the even sample currently being modified is alternately the sample of rank immediately lower or immediately higher. According to the second aspect of the invention, according to a particular characteristic, the filtering method includes, at the end of the aforementioned iteration, an additional filtering step including an operation of weighting by means of a sixth weighting coefficient. According to a particular characteristic, the sixth weighting coefficient is a function of the weighting coefficient used at the previous step, as follows:
In the application of the invention to the filtering of digital images, the output digital signal represents an image. According to a particular characteristic of the first and second aspects of the invention, the modification operations consist of applying an approximation function. This function may be: the identity function, or a function of a real variable which supplies the integer closest to the variable, or a function of a real variable which supplies the first integer below the variable, or a function of a real variable which supplies the first integer above the variable, or a function of a variable decomposed into subvariables whose sum is equal to the variable, which supplies a sum of approximate values of the subvariables, each of the approximate values of the subvariables being either a function of a real variable which supplies the integer closest to the variable, or a function of a real variable which supplies the first integer below the variable, or a function of a real variable which supplies the first integer above the variable. The invention also relates to a signal processing device which has means adapted to implement a filtering method as above. For the same purpose as indicated above, the present invention also proposes, according to a third aspect, a digital filtering device adapted to transform an input digital signal into one or more output digital signals containing even-indexed samples and odd-indexed samples, this filter having at least one weighting module, a module for modifying even-indexed samples by means of a function of weighted odd-indexed samples, a module for modifying odd-indexed samples by means of a function of weighted even-indexed samples, these weighted samples being supplied by the weighting module, the modification modules functioning iteratively, so as to modify even-indexed samples at least once and then odd-indexed samples at least once, this filtering device being remarkable in that at least one of the aforementioned weighting modules receives as an input the difference between two consecutive even-indexed samples. For the same purpose as indicated above, the present invention also proposes, according to a fourth aspect, a digital filtering device adapted to transform one or more input digital signals into an output digital signal, the input signals containing even-indexed samples and odd-indexed samples, this filtering device having at least one weighting module, a module for modifying odd-indexed samples by means of a function of weighted even-indexed samples, a module for modifying even-indexed samples by means of a function of weighted odd-indexed samples, the weighted samples being supplied by the weighting module, the modification module functioning iteratively, so as to modify odd-indexed samples at least once and then even-indexed samples at least once, this filtering device being remarkable in that at least one of the aforementioned weighting modules receives as an input the difference between two consecutive even-indexed samples. The particular characteristics and the advantages of the filtering device being similar to those of the filtering method according to the invention, they are not stated here. Still with the same purpose, the invention also proposes a signal processing device including a filtering device as previously defined, or the means of implementing the corresponding method. The invention also concerns a signal processing device including at least two filtering devices as previously defined, the output signal of one of the filtering devices being the input signal of the other filtering device. The invention also concerns a digital apparatus including the signal processing device. The invention also concerns a digital photographic apparatus including the signal processing device. Still for the same purpose, the present invention also proposes an encoding method including steps adapted to implement a filtering method as above. Still for the same purpose, the present invention also proposes an encoding device including at least one filtering device as above, or the means of implementing the corresponding method. Still for the same purpose, the present invention also proposes a digital signal compression method, including steps adapted to implement a filtering method as above. Still for the same purpose, the present invention also proposes a digital signal compression device, including at least one filtering device as above, or the means of implementing the corresponding method. An information storage means which can be read by a computer or by a microprocessor, integrated or not into the device, possibly removable, stores a program implementing a filtering method as above. The present invention also relates to a computer program product containing sequences of instructions for implementing a filtering method as above. As the particular characteristics and the advantages of the signal processing devices, of the digital apparatus, of the digital photographic apparatus, of the encoding and compression devices and methods, of the storage means and of the computer program product are similar to those of the filtering method according to the invention, they are not stated here. Other aspects and advantages of the invention will emerge from a reading of the following detailed description of a particular embodiment, given by way of non-limitative example. The description refers to the drawings which accompany it. According to a chosen embodiment depicted in The source It will be considered more particularly hereinafter that the data to be encoded are a series of digital samples representing an image IM. The source Means The user means The encoding device The transformation circuit The circuit Means Means The decoding device The circuit The encoding device and the decoding device can be integrated into the same digital apparatus, for example a digital camera. In this case, the data processing device effects the encoding and the decoding of the data. With reference to The device a central processing unit a read-only memory a random access memory a screen a keyboard a hard disk a disk drive an interface an input/output card The random access memory a register “k”, containing the current value of the variable k defined below, a register “j”, containing the current value of the variable j defined below, a register “y a register “β a register “m a register “γ”, in which there are stored the values of the coefficients γ defined below. In the case where the operating program of the central processing unit the maximum value k the value of the parameters L the values of the weighting coefficients α It should be noted that, in the case where The central processing unit The hard disk In more general terms, the programs according to the present invention are stored in a storage means. This storage means can be read by a computer or by a microprocessor. This storage means is integrated or not into the device, and may be removable. For example, it may include a magnetic tape, a diskette or a CD-ROM (fixed-memory compact disc). The device The device The device The screen With reference to In general terms, the resolution of a signal is the number of samples per unit length used for representing this signal. In the case of an image signal, the resolution of a sub-band signal is related to the number of samples per unit length used for representing this sub-band signal horizontally and vertically. The resolution depends on the number of decimations carried out, on the decimation factor and on the resolution of the initial image. The first analysis unit B The sub-band signal LL Each sub-band signal is a set of real coefficients constructed from the original image, which contains information corresponding to an orientation which is respectively vertical, horizontal and diagonal of the contours of the image, in a given frequency band. Each sub-band signal can be assimilated to an image. The sub-band signal LL Each of the sub-band signals of resolution RES To a given analysis circuit An input signal x An initialisation step During step During step If a signal y Thus a signal extension procedure (not depicted in The example, in no way limitative, of an extension procedure given above applies to signals such that the rows i The following step In a preferred embodiment, this lifting step is either of the LSC or of the RSC type, according to the value of the parameter m This lifting step corresponds to the HSA high-pass filter H At the end of the lifting step As long as the result of the test This is because, during the step The non-zero parameter α Given that the value of m The single coefficient low-pass lifting step is either of the LSC or RSC type, according to the value of the parameter m There is a systematic alternation of the following single coefficient low-pass lifting steps (that is to say corresponding to following values of j): for example, the second is of the LSC type if the first is of the RSC type or vice versa. The last single coefficient low-pass lifting step is always of the LSC type. The low-pass lifting step of equation (17) guarantees that the support of the equivalent low-pass filter increases until it reaches the value of the support of the equivalent high-pass filter of the previous high-pass lifting step, i.e. the closest higher even value. In accordance with a preferred embodiment of the present invention, the high-pass lifting step of equation (18) is of the HSA type, unlike the prior art, which uses solely single coefficient lifting steps and WSA lifting steps for the implementation by lifting of HSS/HSA wavelet filters. The high-pass lifting step of equation (18) guarantees that the support of the equivalent high-pass filter increases by a value of two (that is to say until it reaches the closest higher even value) with respect to the support of the equivalent high-pass filter of the previous high-pass lifting step. The equivalent high-pass filter of the high-pass lifting step of equation (18) is of the HSA type if and only if the equivalent high-pass filter of the previous high-pass lifting step is HSA and the non-zero parameter β
Unlike the low-pass lifting step of equation (17), which lifts the low-pass coefficients from the high-pass coefficients alternately to left and right, the high-pass lifting step of equation (18) lifts the high-pass coefficients from both the left-side and right-side low-pass coefficients. When the result of test In a preferred embodiment, the coefficient γ is calculated according to the weighting coefficient used at the previous high-pass lifting step:
The low-pass lifting step In this preferred embodiment, there is obtained, at the end of step Next an iteration is carried out of steps of WSA low-pass (step For this purpose, following step Steps The WSA low-pass lifting steps
These steps are known per se. Their effect is an increase in the size of the corresponding filter (low-pass for step In order to produce wavelet filters with a unity direct current gain and a Nyquist gain of 2, that is to say with a so-called (1,2) normalisation, no specific normalisation step of the type y The approximation function R is not necessarily the same for all the lifting steps for which it is used. The parameters to be chosen for the lifting implementation of the HSS/HSA filters are L An input signal y An initialisation step All the steps of the reverse lifting implementation are the operations which are the reverse of the lifting steps performed in forward transformation. Moreover, the steps of the reverse lifting implementation are performed in the opposite order to the steps of the forward lifting implementation. For an analysis processing followed by a synthesis processing of the same signals, all the parameters of the reverse transformation are identical to those chosen for the forward transformation. As shown by If a signal x Thus a signal extension procedure (not depicted in as defined by equation (24) below for values of i such that mod(i−i as defined by equation (25) below for even values of i such that mod(i−i and as defined by equation (26) below for odd values of i such that mod(i−i There are other extension procedures for other types of filter, such as for example the so-called circular extension, known to a person skilled in the art, for orthogonal filters. In order to implement wavelet filters with a unity DC gain and a Nyquist gain of 2, that is to say with a so-called (1,2) normalisation, no specific normalisation step of the type x As shown in For this purpose, the value of the counter k is decremented by one unit after each passage through step Steps The WSA low-pass lifting steps
These steps are known per se. When the result of test In a preferred embodiment, the coefficient γ is calculated as a function of the weighting coefficient used at the following high-pass lifting step:
During step At the end of the low-pass lifting step As long as the result of test This is because, during step When the result of test As shown by Then a step If the result of test At step If the result of test If the result of test At step If the result of test If the result of test At the end of step At step At the end of step During step In summary: when l when l when l These three procedures are now described in more detail. As shown by Then a step If the result of test If the result of test At the end of step Then this step As shown by Then a step If the result of test If the result of test At the end of step Then this step As shown in The following step At the end of step Then a step If the result of test If the result of test If the result of test During step Then this step If the result of test During step Then this step The filtering device It has an input E The input E The input E The first decimator D The assembly formed by the filter A The output of the second decimator D Consequently the adder AD The output of the adder AD The assembly formed by the filter A The output of the first decimator D The output of the adder AD The adder AD The filtering device This transformation unit has a first input E The signals to be transformed contain here the samples obtained after analysis filtering of a digital signal by the analysis unit of More precisely, the first signal to be transformed contains the low-frequency samples {y The synthesis unit has a structure similar to that of the analysis unit, and can be derived from it simply. In particular, the synthesis unit uses the same filters A The input E The input E The assembly formed by the filter A The output of the subtracter SO The output of the subtracter SO The assembly formed by the filter A The output of the subtracter SO The output of the subtracter SO The output of the subtracter SO An example, in no way limitative, of forward and reverse transformation by lifting for the implementation of wavelet filters according to the invention is now given. The pair of filters used in this example are as follows:
The lifting parameters corresponding to this pair of filters are:
An approximation function R identical for all the lifting steps, defined by R(x)=E(x+½) where E designates the integer part, is also chosen. The length of the signal is chosen so as to be equal to 6: i With regard to the forward lifting equations, all the values of the output signal Y are initialised to the values of the input signal X. The signal Y is then extended by one sample at each limit: y The following four lifting operations are performed. First of all, the values of the samples (y Next, the values of the samples (y Then the values of the samples (y Finally, the values of the samples (y With regard to the reverse lifting equations, all the values of the output signal X are initialised to the values of the input signal Y. The signal X is then extended by two samples at each limit: x The following four reverse lifting operations are performed. First of all, the values of the samples (x Next, the values of the samples (x Then the values of the samples (x Finally, the values of the samples (x The field of application of the present invention is not limited to HSS/HSA wavelet filters with equal supports, but extends much more broadly, on the one hand, to orthogonal wavelet filters and, on the other hand, to any wavelet filters with equal supports. With regard to the application of the invention to the implementation by lifting of orthogonal wavelet filters, the only difference compared with what was described previously lies in the value of the lifting parameters α All the orthogonal filters have equal support and even length but are not symmetrical, except in one trivial case: the Haar filter, defined by H The orthogonal filters of length 4 are characterised by a parameter θ representing one degree of freedom: In more general terms, the orthogonal filters satisfy the equation:
In the non-limitative example given here, as before, an approximation function R identical for all the lifting steps is chosen and defined by R(x)=x+½. The length of the signal is chosen so as to be equal to 6: i The implementation by lifting of such a filter is carried out using the same steps as in the previous example (except with regard to the extension procedure, described below) except that the parameters are here chosen as follows:
The lifting factorisation procedure given above is similar for orthogonal filters. The only difference lies in the relationship between the coefficients α As has been seen for HSS/HSA filters, the parameters α As has been seen for the HSS/HSA filters, the parameters β Finally, as has been seen for the HSS/HSA filters, the parameter γ is a function of the parameters β For transformation by forward lifting, all the values of the output signal Y are initialised to the values of the input signal X. The signal Y is then extended by one sample at each limit: in the example supplied here, use is made of an extension procedure known to a person skilled in the art as circular extension: y All the subsequent steps are identical to those of the previous example, using the values given in equation (52). For transformation by reverse lifting, all the values of the output signal X are initialised to the values of the input signal Y. The signal X is then extended by two samples at each limit: x All the subsequent steps are identical to those of the previous example, using the values given in equation (52). With regard to the application of the invention to the implementation by lifting of any wavelet filters with equal support, the only differences compared with what was described above for the HSS/HSA filters and the orthogonal filters are as follows: there is no dependency relationship between the parameters α the first high-pass lifting step in the forward implementation by lifting can be any, that is to say: It should be noted that, for example, if use is made, in the forward transformation, instead of equation (16), of the following equation:
All the lifting coefficients α Patent Citations
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